A reliability-based maintenance technicians' workloads optimisation model with stochastic consideration

Ighravwe, D. E.; Oke, S. A.; Adebiyi, K. A. Article A reliability-based maintenance technicians' workloads optimisation model with stochastic consideration Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Ighravwe, D. E.; Oke, S. A.; Adebiyi, K. A. (2016) : A reliability-based maintenance technicians' workloads optimisation model with stochastic consideration, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 12, pp. 171-183, https://doi.org/10.1007/s40092-015-0134-6 This Version is available at: https://hdl.handle.net/10419/157477 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL RESEARCH A reliability-based maintenance technicians’ workloads optimisation model with stochastic consideration D. E. Ighravwe 1,2 •S. A. Oke 1 •K. A. Adebiyi 2 Received: 27 November 2014 / Accepted: 5 December 2015 / Published online: 18 December 2015 ÓThe Author(s) 2015. This article is published with open access at Springerlink.com Abstract The growing interest in technicians’ workloads research is probably associated with the recent surge in competition. This was prompted by unprecedented technological development that triggers changes in customer tastes and preferences for industrial goods. In a quest for business improvement, this worldwide intense competition in industries has stimulated theories and practical frameworks that seek to optimise performance in workplaces. In line with this drive, the present paper proposes an optimisation model which considers technicians’ reliability that complements factory information obtained. The information used emerged from technicians’ productivity and earned-values using the concept of multi-objective modelling approach. Since technicians are expected to carry out routine and stochastic maintenance work, we consider these workloads as constraints. The influence of training, fatigue and experiential knowledge of technicians on workload management was considered. These workloads were combined with maintenance policy in optimising reliability, productivity and earned-values using the goal programming approach. Practical datasets were utilised in studying the applicability of the proposed model in practice. It was observed that our model was able to generate information that practicing maintenance engineers can apply in making more informed decisions on technicians’ management. Keywords Experiential knowledge Stochastic workloads Goal programming Technician’s reliability  Technician’s fatigue Introduction Maintenance workload optimisation scheduling problem has witnessed the application of different modelling approaches in improving equipment availability (Safaei et al. 2008). Performance metrics, methodologies and tools used to ascertain the control of technicians’ workload during the plant’s operating hours have evolved for decades. The pressure and growing interest in technicians’ workload practices and research seems to be associated strongly with the recent surge in competition globally. It was also triggered by the increasingly difficulty in economic growth of industries. Current research trends in this area therefore match technicians’ characteristics with workloads and other considerations. There are immense benefits in understanding the reliability of technicians (Gregoriades and Sutcliffe 2008). This will help in the integration of technicians’ reliability into technicians’ work performance evaluation. All these efforts are to minimise maintenance job rework and reduce overtime. It also aids making the overhaul activity more effective and reduce equipment breakdown to an acceptable level. This study is motivated by lack of information on how to optimise the technicians’ costs, workloads, sizes and reliability of technicians. Most theoretical accounts of reliability have been skewed towards machines. An organisation with an effective and efficient group of technicians experiences several benefits. These benefits are &S. A. Oke [email protected] 1 Department of Mechanical Engineering, Faculty of Engineering, University of Lagos, Room 10, Mezzanine Complex, Akoka-Yaba, Lagos, Nigeria 2 Department of Mechanical Engineering, Faculty of Engineering, Ladoke Akintola University of Technology, Ogbomosho, Nigeria 123 J Ind Eng Int (2016) 12:171–183 DOI 10.1007/s40092-015-0134-6 improved productivity, elevated sales, enhanced customer service, reduced labour costs, improved employee engagement and satisfaction (Bartels and Richey 2008). Focusing on the maintenance department as a major area for driving the above-mentioned benefits is important because of its interrelationships with other departments. Maintenance problems in manufacturing systems are diverse. Technicians’ workload and reliability problems, which are key problems in solving other problems in maintenance system, are least considered in literature. Several contributions on spare parts management, machine availability (Safaei et al. 2008) and maintenance budget (Mansour 2011; Ighravwe and Oke 2014; Ighravwe et al. 2015) have been documented. Within the last few decades, researchers and industrial practitioners have started considering human factors in maintenance systems with a view of ensuring robust maintenance management systems. Dealing with the maximisation of a multi-objective problem that is nonlinear in formulation, a single objective model with weights for each objective was considered. Such a model can be solved easily using metaheuristics (evolutionary algorithms and swarm algorithms) in order to generate compromise solutions. Currently, there exists sparse documentation on big-bang–big-crunch (BB–BC) and EP algorithms as solution methods for technicians’ parameters optimisation. Studying fatigue during the design of technicians’ workload and reliability for planned operations will help in improving work-plans. Another aspect of technician’s problem, which has been sparingly considered in the literature, is experiential knowledge of the technicians who engage in rework activities. The objective of this study was to develop a mixedinteger optimisation model that optimises technicians’ sizes, workloads (service times), reliability, availability, performance (efficiency) and quality of workdone. A performance comparative analysis of EP and BB–BC algorithms when used in solving the proposed model was also carried out. The selection of these algorithms was motivated by their low computation time. Literature review Maintenance system operations are often executed using maintenance teams (Hedjazi 2015). These teams are formed based on the skills of technicians (Hedjazi 2015) and the types of maintenance activities (Ighravwe and Oke 2014; Ighravwe et al. 2015) in an organisation. Tohidi and Tarokh (2005) reported that when forming teams, there is the need to ensure proper communication systems. These help in improving the productivity of teams. Thus, the interest of researchers and practitioners in maintenance systems is on the analysis of types of maintenance activities, policies, technicians’ attributes (productivity, service time and skills) and cost of operating maintenance systems (Lai et al. 2015). The core reason for studying maintenance departmental performance is to improve organisation’s profitability (Alsyouf 2007). This has stimulated different scientific methods in analysing maintenance systems. Maintenance workload management helps in determining equipment downtime. When workload is optimally shared among the team members, low amounts of equipment downtime will be experienced. For example, the concern of Oladokun et al. (2006) was how to predict equipment downtime. In their study, a predictive model was designed to predict equipment downtime using machine breakdown periods and maintenance repair time as dependent variable. The issue of how to determine maintenance repair times was not exhaustively addressed in their study. Rana and Purohit (2012) investigated the application of critical path method in addressing the problems of technician’s productivity, maintenance time and tasks balancing. One drawback of their study is the assumption that technicians are capable of doing any maintenance work in an organisation. In practice, this assumption is feasible for small-scale organisations. However, for large-scale systems, the assumption may be violated. This study relaxed the assumption of Rana and Purohit (2012) by considering technicians for different sections (mechanical, electrical and instrumentation). The scope of Ighravwe and Oke’s (2014) study limited maintenance activities to routine maintenance. Ighravwe and Oke (2014) modelled maintenance time utilisation problem by considering the allocated and actual maintenance time used by technicians. No mention of technician’s training and fatigue experienced during the execution of maintenance activities was mentioned in their work. In the work of He et al. (2014), the issue of technician’s reliability and cost were addressed using nonlinear programming techniques. Their proposed model can be applied in a maintenance system. However, consideration was not given to the impact of technician’s fatigue, experiential knowledge and training on technician’s reliability. Huge amount of funds are usually invested in maintenance systems of organisations. This is necessary avert machine breakdowns, which could be more costly. Fajardo and Drekic (2015) pointed out that cost-effective maintenance checks could be achieved during close-down periods. The problem of technician cost and equipment availability was studied by Safaei et al. (2008). The application of simulated annealing as a solution method for handling multi-objective technicians’ problem was demonstrated. They considered the management of regular, outsourcing and overtime maintenance activities. Mansour’s (2011) studied maintenance cost minimisation 172 J Ind Eng Int (2016) 12:171–183 123 problem by proposing a mixed-integer programming model that incorporates equipment complexity. Their work investigated the suitability of genetic algorithm as a solution method for technicians’ parametric optimisation. Kaufman and Lewis (2007) considered the application of repair and replacement models for maintenance workload management. In their study, information on service rates and failures, fixed maintenance times and random replacement times was generated. A model that minimised the total cost of technicians used for maintenance activities was developed by De Bruecker et al. (2015) using mixedinteger programming approach. In their study, a model enhancement heuristic was proposed as solution method for their model. The heuristic considered stochastic service levels. Knapp and Mahajan (1998) conducted a comparative analysis between centralised and decentralised organisational structure as it affects maintenance systems. Consideration was given to technician-type (in-house and subcontracted technicians) and training levels of technicians. The results of their study demonstrated how optimal allocation of technicians can be achieved. A study which considered organisational policies on technicians’ capacity was presented by Mjema (2002). The challenge of managing the switches of technicians for mechanical and electrical workloads was considered. Their analysis was based on the technicians’ utilisation and through-put time, work-order requirement and prioritisation rules. Manzini et al. (2015) presented a nonlinear model which minimises cost of preventive and corrective maintenance as well as technicians’ workload and spare parts cost. These maintenance costs were optimised under total expected and probabilistic costs. Hervet and Chardy (2012) reported the use of mixed-integer programming approach in addressing the problem of preventive maintenance workforce needs during the design of passive optical network. Jarugumill (2011) studied workforce problem by considering regular and overtime activities as well as workers’ skills. Research methodology This section presents the proposed model (‘‘Model formulation’’ section) and discussed the solution methods used in solving the proposed model (‘‘Solution methods’’ section). The development of the proposed model was based on the following assumptions: 1. Technicians in the same group can be categorised differently, and these categories are known in advance; 2. Available maintenance work is carried out by in-house maintenance crew only; and 3. Inventories for maintenance work are available and released when needed. Some of the notations used in formulating the proposed model are given as follows: Indices iMaintenance activity jMaintenance section kTechnicians category tPlanning period MTotal number of types of maintenance activities NTotal number of maintenance sections KTotal number of technicians categories TTotal number of sub-planning periods Decision variables x ijkt Number of technicians required for maintenance activity ifrom maintenance section jbelonging to technician category kat period t R ijkt Reliability of a technician required for maintenance activity ifrom maintenance section jbelonging to technician category kat period t d ijkt Amount of maintenance time required by technician to carry out maintenance activity ifrom maintenance section jbelonging to technician category kat period t(h) Parameters v ijkt Earned-value of a technician that carries out maintenance activity ifrom maintenance section jbelonging to technician category kat period t(N) c ijkt Unit cost of technician that carries out maintenance activity ifrom maintenance section jbelonging to technician category kat period t(N) r it Expected value of technicians’ reliability for maintenance activity iat period t vit Total cost of technicians required to carry out maintenance activity iat period t(N) vtTotal cost of technicians required to carry out available maintenance activities at period t(N) a ijt Actual number of days a technician in section jfrom technicians category kat period tis available in a maintenance system (days) p ijt Actual performance of a technician in section jfrom technician’s category kat period t q ijt Actual quality of workdone by a technician in section jfrom technician’s category kat period t(kg) J Ind Eng Int (2016) 12:171–183 173 123 ^ aijt Expected days a technician in section jfrom technician’s category kat period tis available at a maintenance system (days) ^ pijt Expected performance of a worker in section jfrom technician’s category kat period t ^ qijt Expected quality of workdone by a technicians in section jfrom technician’s category kat period t(kg) bijk 1The minimum value of fatigue experienced during maintenance activity iby a technician from maintenance section jbelonging to technicians’ category k(h) bijk 2The maximum value of fatigue experienced during maintenance activity iby a technician from maintenance section jbelonging to technicians’ category k(h) cjk 1The minimum value of extra fatigue on technicians for overtime activities by technicians from maintenance section jbelonging to technician’s category k(h) cjk 2The maximum value of extra fatigue on technicians for overtime activities by technicians from maintenance section jbelonging to technician’s category k(h) ejk 1The minimum value of experiential on technicians for reworked activities by technicians from maintenance section jbelonging to technician’s category k(h) ejk 2The maximum value of experiential on technicians for reworked activities by technicians from maintenance section jbelonging to technician’s category k(h) Vijk 2The minimum value of training impact on technicians for maintenance activity iby technicians from maintenance section jbelonging to technician’s category k(h) Vijk 2The maximum value of training impact on technicians for maintenance activity iby technicians from maintenance section jbelonging to technician’s category k(h) fbðÞ Probability density function of fatigue on technicians for maintenance activities fvðÞ Probability density function of training impact on technicians service times feðÞ Probability density function of extra fatigue experience by technicians during overtime maintenance activity fcðÞ Probability density function of experience gain by technicians during reworked maintenance activity Model formulation This study presented two technicians’ objectives (earnedvalue and reliability) which were subjected to technician’s cost, service time, reliability, availability, performance and quality of workdone. The cost of keeping a particular level of technician in a system depends largely on the expected contributions from the technicians. Using the concept of earned-value from engaging technicians in planned and unplanned maintenance activities in an organisation, the total earned-value from scheduling the different technicians for stochastic maintenance activities in a maintenance system was expressed as Eq. (1). The technician’s earned-value is a function of the size of technicians assigned to carry out maintenance activities ðxijktÞ;time spent on maintenance activities ðdijktÞand the unit earned-value expected from each technician ðvijktÞ:Technician’s earned-value could be defined as a measure of the value of utilising a technician for maintenance activities with respect to maintenance time. Max f1¼X T t¼1X M i¼1X N j¼1X K K¼1 vijktdijktxijkt ð1Þ By looking beyond the concept of earned-value of technicians, technicians scheduling analysis may be improved on using the average expected technician reliability ðRijkÞover a planning period. The study of technician’s reliability provides a means of estimating the degree to which the expected earned-value of technicians will be achieved. Since technicians work in groups, we considered their reliability as being parallel. The expected average technicians’ reliability is expressed as Eq. (2). Max f2¼1 TX T t¼1Y M i¼1 11Y N j¼1Y K k¼1 11Rijkt  xijkt  ! !"# ð2Þ The volume of maintenance work from overtime, overtime and rework maintenance activities in a system varies from one period to another. This variation may be attributed to equipment usage, equipment age, organisation’s maintenance policy and quantity of spare parts used. Furthermore, factors such as fatigue, training and experiential knowledge affect technicians performance when restoring equipment to acceptable functional state. The issue of routine maintenance activities has been studied in the literature (Mansour 2011). However, the concerns of researchers have been on equipment. Information on factors which affect technicians’ performance is sparse in maintenance literature. This study considered the issue of fatigue and training on technician’s service time. Since it is often difficult to measure fatigue in qualitative terms, a continuous function was used to capture the amounts of fatigue a technician experience during the maintenance activities. This is possible by considering two 174 J Ind Eng Int (2016) 12:171–183 123 extreme values for technician’s fatigue for a particular kind of maintenance activity. The expected reduction in maintenance time of technicians in a section is expressed as Eq. (3). where L1jt and  L1jt are the minimum and maximum amounts of routine maintenance tasks available for maintenance section jat period t, respectively. The extension of production activities beyond the scheduled periods makes it necessary for technicians to be on ground. The technicians are expected to carry out maintenance activities that are required during overtime production activities. The need for overtime activities may be attributed to a change in production volume and maintenance-related problems. Shiftan and Wilson (1994) considered this problem under a deterministic condition. This study deviated from Shiftan and Wilson (1994) approach by introducing stochastic element into overtime maintenance activities. These stochastic elements involve the fatigue experienced during normal and overtime periods as well as reduction in maintenance time as a result of training technicians (Eq. 4). The determination of the number of overtime technicians will improve technicians’ cost control in systems (Aghdaghi and Jolai 2008). where L2jt and  L2jt are the minimum and maximum amounts of overtime maintenance tasks available for maintenance section jat period t, respectively. The failures of machines to produce the expected number of defective products after maintenance often result in remaintenance (rework) of such machines. This problem results from poor diagnosis of the causes of breakdown of installed machines or the use of inferior spare parts. Whenever this problem occurs, technicians who are responsible for the initial maintenance of such machines could have gained experience on the possible causes of the poor machine performance. By the combining experiential knowledge, training and fatigue, the constraint for the expected time for sectional rework activities is given as Eq. (5). where L3jt and  L3jt are the minimum and maximum amounts of rework maintenance tasks available for maintenance section jat period t, respectively. L1jt P K k¼1 d1jkt R bijk 2 bijk 1 bfbðÞdb ! x1jkt   L1jt t¼1 L1jt P K k¼1 d1jkðt1ÞþR vijk 2 vijk 1 vfvðÞdvR bijk 2 bijk 1 bfbðÞdb ! ! x1jkt   L1jt Otherwise 8 > > > > < > > > > : 8j;tðÞ ð3Þ L2jt P K k¼1 d2jkt R bijk 2 bijk 1 bfbðÞdbR cjk 2 cjk 1 cfcðÞdc ! x2jkt   L2jt t¼1 L2jt P K k¼1 d2jkðt1ÞþR vijk 2 vijk 1 vfvðÞdvR bijk 2 bijk 1 bfbðÞdbR cjk2 cjk 1 cfcðÞdc ! x2jt   L2jkt Otherwise 8 > > > > < > > > > : 8j;tðÞ ð4Þ L3jt P K k¼1 d3jkt þR ejk 2 ejk 1 efeðÞdeR bijk 2 bijk 1 bfbðÞdb ! x3jkt   L3jt t¼1 L3jt P K k¼1 d3jkðt1ÞþR vijk 2 vijk 1 vfvðÞdvþR ejk 2 ejk 1 efeðÞdeR bijk 2 bijk 1 bfbðÞdb ! x3jkt   L3jt Otherwise 8 > > > > < > > > > : 8j;tðÞ ð5Þ J Ind Eng Int (2016) 12:171–183 175 123 Using the expected value and standard deviation of technician in a unit for the classes of technicians in each section, the minimum number of technicians required for each class under the various units in a maintenance department can be expressed as follows: dijktxijkt lxijkt ar 1 NX N i¼1 x2 ijktPx ijkt  2lxijkt X N i¼1 x2 ijktPx ijk  þl2 xijkt X N i¼1 Px ijkt  !!1=2 8i;j;k;tðÞ ð6Þ where lxijkt is mean value of maintenance task ifrom maintenance section jfor technician kat period t.PðxijktÞis the probability of occurrence of maintenance task ifrom maintenance section jfor technician kat period tand ar contribution of variance in maintenance tasks to lxijkt : The average number of technicians required in a planning horizon should not be less than the specified number of technicians (Eq. 7). This allows variations in the number of technicians in a section from one period to another. PT t¼1xijkt Txijk;ave 8i;j;kðÞ ð7Þ where xijk;ave is the average number of technicians expected to be scheduled to carried out maintenance task ifrom maintenance section jbelonging to technician’s category iat period t. The sum of technicians in each period is expected to be within a specified range. This constraint helped in controlling the amount of technicians within a particular category (Eq. 8). xxmin X M i¼1X N j¼1X K k¼1 xijkt xxmax 8tð8Þ In Ighravwe et al. (2015), technicians’ overall effectiveness (TOE) was considered. To control the expected value of TOE, there is the need to consider constraining the minimum values for sectional technicians’ availability, quality of workdone and performance. The expected performance of each section in a maintenance department could be considered as the expected technician’s efficiency. We applied uniform distribution concept (Wu 2008)to estimate the expected technicians’ availability, quality of workdone and performance. Technicians’ availability is expressed as Eq. (9), while technicians’ performance was expressed as Eq. (10). The quality of workdone constraints is expressed as Eq. (11). PK k¼1aijktxijkt PK k¼1 ^ aijktxijkt   bj aj  1 aj  þ aj8ði;j;tÞð9Þ PK k¼1pijktxijkt PK k¼1 ^ pijktxijkt b _ ja _ j  1a _ j  þa _ j8ði;j;tÞð10Þ PK k¼1qijktxijkt PK k¼1 ^ qijktxijkt  ~ bj~ aj  1~ aj  þ~ aj8ði;j;tÞð11Þ where a _ jand b _ jare the minimum and maximum values of technicians’ performance for maintenance task ifrom technicians in section jat period t, respectively.  ajand  bj are the minimum and maximum values of technicians’ availability for maintenance task ifrom technicians in section jat period t, respectively. ~ ajand ~ bjare the minimum and maximum values of technicians’ quality of work for maintenance task ifrom technicians in section jat period t, respectively.  aj,a _ jand ~ ajare the confident levels for technicians’ availability, performance and quality of workdone for maintenance task iexpected from technicians in section j. We relaxed the expression for TOE as defined by Ighravwe et al. (2015), to a stochastic constraint using the concept of normal distribution (Wu 2008) as Eq. (12). PN j¼1PK k¼1aijktxijkt PN j¼1PK k¼1 ^ aijktxijkt PN j¼1PK k¼1pijktxijkt PN j¼1PK k¼1 ^ pijktxijkt PN j¼1PK k¼1qijktxijkt PN j¼1PK k¼1 ^ qijktxijkt liþU11ai ðÞri8ði;tÞ ð12Þ where li;riand aiare the mean, standard deviation and confident level for TOE expected for maintenance task ifrom the technicians in a maintenance system, respectively. Beyond specifying the expected TOE of technicians, there is the need for the cost implication consideration that will be associated with scheduling a particular level of technicians for each maintenance task at the different planning periods. The expression of the cost for each maintenance activity is considered (Eq. 13). The total cost of technicians required for the maintenance activities at each period is expressed as Eq. (14). X N j¼1X K K¼1 Cijktdijktxijkt vit 8ði;tÞð13Þ X M i¼1X N j¼1X K K¼1 Cijktdijktxijkt  vt8ðtÞð14Þ 176 J Ind Eng Int (2016) 12:171–183 123 To further constraint the proposed model, the issue of the expected reliability from each maintenance section is considered. The reliability of each section in a maintenance department is expressed as Eq. (15). 1Y N j¼1Y K k¼1 11Rijk  xijkt  ! rit 8ði;tÞð15Þ Solution methods The handling of the multi-objective is carried out using the weights of each of the objectives and deviational variables for the objective functions (Wu 2008). The new objective function is given as Eq. (16). Fig. 1 Flow chart for the solution methods (Wong and Yuryevich 1997; Sakthivel and Mary 2013) J Ind Eng Int (2016) 12:171–183 177 123 Minimise w1dþ 1þd 1  f1;max f1 þw2dþ 2þd 2  f2;max f2 ð16Þ where f 1,max and f 2,max are the maximum values for the technicians earned-value and reliability, respectively. The summary of the proposed mixed-integer programming model is presented as follows: Minimise w1dþ 1þd 1  f1;max f1 þw2dþ 2þd 2  f2;max f2 Subject to the following constraints: f1þd 1¼f1;max ð17Þ f2þd 2¼f2;max ð18Þ Equations (3)–(15). Non-negativity constraints The flow chart for the evolutionary programming and bigbang–big-crunch algorithms is depicted in Fig. 1. The termination of each of these algorithms was taken as the maximum epoch. EP algorithms are different from other evolutionary algorithms (genetic algorithm, differential evolution and genetic programming) because it does not require crossover operation. The first EP algorithm was proposed by Fogel (1962). The quest to improve the mutation operation in EP algorithms has led to different versions of the EP algorithms. The potentials of EP and BB–BC algorithms in generating optimal solution for computational problems are due to their stochastic-populate-based capacity. For the EP algorithm, the mutation introduced randomness into a current solution. In the BB–BC algorithm (Osman and Eksin 2006), randomness to current solution is introduced through the generation of centre of mass (bigcrunch) and the new variable (big-bang). In Fig. 1,zg ij is the value of solution idecision variable jat epoch gand zg jthe centre of mass for decision variable jat epoch g. The variable zgj is the value of global solution variable j in a current epoch. Some authors have considered zgj as the centre of mass during the implementation of BB–BC algorithm. The variable fg iis the quality of solution iat epoch g. In the EP section, w1is a uniform random number which lies between (0, 1). This variable helps in controlling the influence of the difference between current global and local optimal solutions in a population at a particular epoch g.Thevariablew2in the BB–BC algorithm is a random variable which lies at ±1, and # is a constant parameter that helps in controlling the search capacity of the BB–BC algorithm. The variables zj;min and zj;are the minimum and maximum values of decision variable j. Model application The proposed model and algorithms (BB–BC and EP) were coded using C# programming language on a Windows 8 computer with installed memory (RAM) of 4.00 GB, 1.80 GHz processor and 64 bit operating system. To demonstrate the applicability of the proposed model, datasets from Ighravwe and Oke (2014) were used and complemented with simulated data. The datasets that were simulated are technician’s reliability, earned-value, quality of workdone, availability and performance as well as the amount of rework and overtime maintenance activities. The amount planned maintenance work in Ighravwe and Oke (2014) was increased by 20 %. This enables us to generate upper bounds for the various planned maintenance activities. The minimum value for the amounts of overtime was about 40 % of minimum planned maintenance workload. We considered the amount of rework maintenance activities as 30 % of the amounts of minimum planned maintenance workload. Table 1shows the amount of planned, rework and overtime maintenance activities for the system. By observing the total amounts of workloads for the different sections, available maintenance time and Table 1 Simulated maintenance time for the different maintenance activities Technician t=1t=2t=3t=4 Planned maintenance (h) x 111t 3132.95 2950.99 3182.18 2889.25 x 112t 2545.74 2490.33 2338.70 2371.24 x 121t 1472.62 1395.36 1389.03 1358.50 x 122t 1082.14 1060.97 1040.93 1152.72 x 131t 581.39 629.68 579.78 627.42 x 132t 592.05 574.70 533.63 545.71 Overtime maintenance (h) x 211t 1253.18 1180.40 1272.87 1155.70 x 212t 1018.30 996.13 935.48 948.50 x 221t 589.05 558.15 555.61 543.40 x 222t 432.85 424.39 416.37 461.09 x 231t 232.56 251.87 231.91 250.97 x 232t 236.82 229.88 213.45 218.29 Rework maintenance (h) x 311t 939.89 885.30 954.65 866.78 x 312t 763.72 747.10 701.61 711.37 x 321t 441.79 418.61 416.71 407.55 x 322t 324.64 318.29 312.28 345.82 x 331t 174.42 188.91 173.94 188.23 x 332t 177.62 172.41 160.09 163.71 178 J Ind Eng Int (2016) 12:171–183 123